Learning and Generating Mixed States Prepared by Shallow Channel Circuits
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arXiv
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| Format: | Preprint |
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2026
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| author | Hu, Fangjun Kokail, Christian Kornjača, Milan Lopes, Pedro L. S. Gong, Weiyuan Wang, Sheng-Tao Gao, Xun Ostermann, Stefan |
| author_facet | Hu, Fangjun Kokail, Christian Kornjača, Milan Lopes, Pedro L. S. Gong, Weiyuan Wang, Sheng-Tao Gao, Xun Ostermann, Stefan |
| contents | Learning quantum states from measurement data is a central problem in quantum information and computational complexity. In this work, we study the problem of learning to generate mixed states on a finite-dimensional lattice. Motivated by recent developments in mixed state phases of matter, we focus on arbitrary states in the trivial phase. A state belongs to the trivial phase if there exists a shallow preparation channel circuit under which local reversibility is preserved throughout the preparation. We prove that any mixed state in this class can be efficiently learned from measurement access alone. Specifically, given copies of an unknown trivial phase mixed state, our algorithm outputs a shallow local channel circuit that approximately generates this state in trace distance. The sample complexity and runtime are polynomial (or quasi-polynomial) in the number of qubits, assuming constant (or polylogarithmic) circuit depth and gate locality. Importantly, the learner is not given the original preparation circuit and relies only on its existence. Our results provide a structural foundation for quantum generative models based on shallow channel circuits. In the classical limit, our framework also inspires an efficient algorithm for classical diffusion models using only a polynomial overhead of training and generation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_01197 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Learning and Generating Mixed States Prepared by Shallow Channel Circuits Hu, Fangjun Kokail, Christian Kornjača, Milan Lopes, Pedro L. S. Gong, Weiyuan Wang, Sheng-Tao Gao, Xun Ostermann, Stefan Quantum Physics Statistical Mechanics Computational Complexity Machine Learning Learning quantum states from measurement data is a central problem in quantum information and computational complexity. In this work, we study the problem of learning to generate mixed states on a finite-dimensional lattice. Motivated by recent developments in mixed state phases of matter, we focus on arbitrary states in the trivial phase. A state belongs to the trivial phase if there exists a shallow preparation channel circuit under which local reversibility is preserved throughout the preparation. We prove that any mixed state in this class can be efficiently learned from measurement access alone. Specifically, given copies of an unknown trivial phase mixed state, our algorithm outputs a shallow local channel circuit that approximately generates this state in trace distance. The sample complexity and runtime are polynomial (or quasi-polynomial) in the number of qubits, assuming constant (or polylogarithmic) circuit depth and gate locality. Importantly, the learner is not given the original preparation circuit and relies only on its existence. Our results provide a structural foundation for quantum generative models based on shallow channel circuits. In the classical limit, our framework also inspires an efficient algorithm for classical diffusion models using only a polynomial overhead of training and generation. |
| title | Learning and Generating Mixed States Prepared by Shallow Channel Circuits |
| topic | Quantum Physics Statistical Mechanics Computational Complexity Machine Learning |
| url | https://arxiv.org/abs/2604.01197 |